international policy coordination in dynamic macroeconomic

This PDF is a selection from an out-of-print volume from the National Bureau of Economic Research

Volume Title: International Economic Policy Coordination

Volume Author/Editor: Buiter, Willem H. and Richard C. Marston, eds.

Volume Publisher: Cambridge University Press

Volume ISBN: 0-521-33780-1

Volume URL: http://www.nber.org/books/buit85-1

Publication Date: 1985

Chapter Title: International Policy Coordination In Dynamic Macroeconomic Models

Chapter Author: Gilles Oudiz, Jeffrey Sachs

Chapter URL: http://www.nber.org/chapters/c4137

Chapter pages in book: (p. 274 - 330)

7 International policy coordination indynamic macroeconomic models

I Introduction

In an earlier essay (Oudiz and Sachs, 1984) we investigated the quantitativegains to international policy coordination in a static environment. In thispaper, we begin to extend the analysis to a dynamic setting. However,because of several new issues, this first step is moretheoretical than empirical. The extension to dynamics introduces threeimportant points of realism to the static game. First, the payoffs tobeggar-thy-neighbor policies may look very different in one-period andmultiperiod games, so that the need for policy coordination may bedifferent in the two games. Second, it is often claimed that governmentsare shortsighted in macroeconomic planning, and support for this view hascome from the literature on political business cycles.' We should thereforeinvestigate whether international policy coordination is likely to exacerbateor meliorate this shortsighted behavior. Third, governments act under afundamental constraint that they cannot bind the actions of later govern-ments (or even of themselves at a future date). In principle, therefore,optimizing governments must take into account how future governmentswill behave in view of the economic environment that they inherit. Westudy the implications for policy coordination of this inability to bindfuture governments.

Let us consider these three points in turn. In the static game, uncoordi-nated macroeconomic policy-making is typically inefficient because of aprisoner's dilemma in policy choices. Consider, for example, two countriesthat are attempting to move optimally along short-run Phillips curves, Itmay be that each country will choose contractionary policies no matterwhat the other country selects, though the policy pair (expand, expand)is better for both countries than the non-cooperative equilibrium (contract,contract). As we showed in our earlier study, this situation arises naturallyunder flexible exchange rates, since by contracting while the other country

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International coordination in dynamic macroeconomics 275

is expanding a country can appreciate its currency and export some of itsinflation abroad. It is this beggar-thy-neighbor action that gives rise to theprisoner's dilemma. Cooperation, say in the form of a binding internationalcommitment to expand, may be useful in moving the countries to theefficient equilibrium.

The question arises whether the payoff structure in a multiperiod, orinfinite-horizon, game will look the same. The reason for doubt is simple.In almost all macroeconomic models, policies which lead to a short-runreal appreciation also lead to long-run real depreciation, or at least a returnto the initial real exchange rate. In this circumstance, farsighted players

understand that a short-run beggar-thy-neighbor appreciation isless attractive than it looks, since it will be reversed in the long run, at whichpoint the country reimports the inflation that it earlier sent abroad. To thisextent, the beggar-thy-neighbor policy loses its appeal, and the need forcoordination is reduced.

The second theme introduced in a multiperiod setting is the myopicr, behavior of governments. In considering public welfare in a multiperiodre game, it is natural to consider a payoff of the form:

7'

to = (1)t.-o

Here, is the intertemporal utility of country i as of time zero. is

ts the instantaneous utility of the country at time t, as function of a vectorof macroeconomic targets T. /1 is a pure rate of time preference, with/i < 1,

re so that the future is discounted relative to the present. The planningte interval is for t between 0 and T.a In view of the evidence on political business cycles, in which governments

attempt to manipulate in conjunction with upcoming elections, it seemse, natural to suggest that if (1) is the 'true' social welfare function, the

government's social welfare function take the form:ed = E /jGtu(Ti) (2)

where TG Tand </1. That is, its planning horizon is shorter than thea economy's, or its discounting of the future is higher.

In this view, the public is partly a hostage of a self-serving government.It The policy choices reflect the incumbent government's goals, and not the

public's. If this is so, we can ask whether international policy coordinationI) is likely to improve or worsen this sub-optimal situation. At an abstract

level, the arguments seem to fall on both sides. Some critics, for example,have characterized policy coordination as a cartel of the incumbents, in

y which each policymaker helps the others to manipulate the political

a

276 Glues Oudiz and Jeffrey Sachs

business cycle. As an example of this, policymakers may have a short-runexpansionary bias if expansion shows up as output today and as inflationonly many years in the future. To some extent, the fear of currencydepreciation following a unilateral expansion keeps this bias in check.That is, the flexible exchange rate provides discipline on the shortsighted ggovernment. With policy coordination, the fear of currency depreciationcan be removed by a commitment of all countries to expand. In this way, rpolicy coordination may give incumbent governments a free hand toundertake overly inflationary policies. I

On the other hand, we can think of circumstances in which policycoordination ties the hand of incumbents, and thus prevents such self-serving policies. An international gold standard, for example, might gimpose discipline on governments that would not exist in each countryalone. To analyse this possibility fully we would have to examine each tigovernment's incentive to stick with a particular rule, and the extent to awhich internationally certified rules are more or less durable than rules tiundertaken unilaterally. For example, each country on its own could adopt e:

a gold standard. What, if anything, is added by a multicouñtry e:commitment?

The third theme introduced in a multiperiod setting is that of 'time-consistency' of optimal plans. Even in circumstances in which the current dgovernment (or current administration) has the public's interest at heart, aits ability to maximize social welfare may be limited by its inability to spre-commit the actions of (well-meaning) future governments. In thesecircumstances, the current government must choose its optimal policy Staking as given the policy rules that will be pursued in the future. That is,it must optimize today, assuming that future governments will optimize Funder the assumption that yet future governments will optimize, and soon. In general this constrained optimization yields a lower level of socialwelfare than does the case in which the government can choose not only Iits own policies but those of future governments as well.

Many authors, including Barro and Gordon (1983) and Rogoff (1983),have given examples in which the inability to bind future policies imparts cdan inflationary bias to the economy. In these examples, wage setters setwages before macroeconomic policy is set. Once the wages are set,policymakers have an incentive to expand the economy to reduce realwages, and raise output. Wage setters anticipate these policies, and chooseinflationary wage settlements in anticipation. If the government can tipre-commit to avoid inflationary policies, the economy can get the same gex post output levels at a lower rate of inflation. Unfortunately, such apre-commitment is not credible since the government has an incentive torenege on it after the wages are set. H

As Rogoff stresses, this time consistency problem may have important l

4


__

consequences for international policy coordination. If the inability to bind

)fl future policies leads to an inflationary bias, international policycoordination may further exacerbate this bias by eliminating each country's

:k. concern about currency depreciation. Thus, even when a sequence ofed governments within each country is trying to maximize that country's true

social welfare function, policy coordination may make the situation worserather than better.

to We consider later on several factors that tend to weaken this pessimisticconclusion. First, in infinite-horizon games, governments may be able to

icy invest in a 'reputation' in order to overcome the time-inconsistencyproblem (as illustrated in Barro and Gordon (1983)). In other words, agovernment's credibility may be judged by its willingness to honor a

• try program laid down by an earlier government, so much that it continues• ich the policy rather than reoptimizing during its incumbency. We will provide

to an example of this solution to the time inconsistency problem. Second, to.les the extent that the time inconsistency problem revolves around the

• )pt exchange rate, policy coordination may actually eliminate the problem. Intry examples later in the paper, optimal coordinated policies in our

two-country model turn out to be time-consistent.The plan of the paper is as follows. In the next section we set out a simple

dynamic macroeconomic model characterized by flexible exchange ratesit, and perfect foresight on the part of the private and public sectors. Into Section III, we describe various equilibria in a one-country version of the

ese model, to highlight the implications of time inconsistency. Next, inicy Section IV, we describe the various equilibria in the two-country version ofis, the game, including the welfare gains or losses from policy coordination.ize Extensions and conclusions are discussed in a final section.

II A simple dynamic macroeconomic model

We consider a simple model of the sort explored by Dornbusch (1976).3), The home country produces output Q, at price P, and trades with a foreignrts country, which produces Q* at price The domestic exchange rate Eset measures units of home currency per unit of foreign currency, so that the

relative price of the home good is P/(EP*). Demand for the home goodeal is a decreasing function of P/(EP*) and of the real interest rate, and an)se increasing function of Q*. Letting lower case variablesp, q, and e representan the logarithms of their upper-case counterparts, we write demand for homerue goods as:

to = (3)

Here, i is the nominal interest rate, and — —ps) the home real interestant rate at time t is the expectation of at time t). Under the perfect

278 Gilles Oudiz and Jeffrey Sachs

foresight assumption, which we hereafter maintain, = for Jr

ecThe money demand equations take the standard transactions form: at

= (4) at01

For convenience, we will invert this equation and write

= (5)

with = and p = 1/c. Following Dornbusch, we assume perfect 01

capital mobility, so that uncovered interest arbitrage holds:,

— = — (6)

Again, assuming perfect foresight, we solve for equilibria with =for all t.

It remains to specify wage and price dynamics. First, the (log) consumerprice index (pC) is written as a weighted average of home (p) and foreign(p*+e) prices

= Apt+(l__A)(pt*+et) (7)

Home prices are written as a fixed markup over wages: cd

(8)

Finally, nominal wage change, — wt, is made a function of laggedconsumer price change, output, and output change: o

= (9)

Note that since Wt+i — is a function of lagged rather than contempor-aneous price change, the system will display typical Keynesian features, uparticularly the of with respect to contemporaneous and dfuture anticipated changes in This is the standard presumption in the TDornbusch model that the labor market clears more slowly than the assetmarkets.

In the next section, we will introduce corresponding equations for thesecond country, in order to construct a two-country model. Here, we focus Nton the one-country case by making the small-country assumption for thehome economy that and q* are given for all t 0. By doing so, wecan write the one-country model as a four-dimensional difference equationsystem as in (l0):2

toPt+i Pt *

PtPt =A Pt-i (10)

* it


In any given period, pj, and are given by the past history of theeconomy. These are the 'pre-determined' variables of the economy, me,and indeed the entire sequence of m, is chosen as a policy variable. p', i',and q' are exogenous forcing variables of the system from the point of view

I) of the home economy.As is typical of perfect foresight models, an asset price such as is

determined not by past history but by forward-looking behavior of asset5) holders. In particular, for given values of and given sequences

of 1*, and q* from t to infinity, there is typically a unique value ofsuch that the exchange rate does not grow or collapse explosively(technically, this unique value of et puts the economy on its stable

5) manifold). Such a unique value of eg exists as long as the eigenvalueassociated with in the A matrix is outside the unit circle, and theremaining eigenvalues are on or within the unit circle. In the simulationsreported below, this condition is always satisfied.

The goal of economic policy in our model will be to maximize a social• welfare function as in (1) or (2), subject to the constraint in (10). The

assumption that eg is always such as to keep the economy on the7) saddlepoint path (or stable manifold) requires that economic agents have

complete knowledge as to the path of future policies. In this sense, thegovernment is like a Stackelberg leader with respect to the private sector,choosing monetary policy with a view to affecting and thereby more basiceconomic targets, while is chosen taking as given the future sequenceof m. This is not to say, however, that governments can necessarily chooseany sequence of en that they desire. A large part of the discussion thatfollows describes the 'admissible' sequences of policies.

As a concrete example of this model, we will suppose that instantaneouss, utility is a quadratic function of inflation, = and the

deviation of output from full employment qt• That is, =Thus, intertemporal utility is

E (11)t—o

Note that çb is a parameter reflecting the weight attached to relative to

fi is the discount factor. We have written the utility function with an'e infinite horizon, and we will point out shortly some special features of then problem that arise with such a formulation.

We now turn to the optimal policy for en. It may seem straightforwardto maximize (11) subject to (10), but as Phelps and Pollak (1968) firstexplained, and Kydland and Prescott (1977) further elucidated, themaximization is quite problematic. Here we sketch the problem, and treatit in greater detail below.

a

280 Gilles Oudiz and Jeffrey Sacha

Suppose that we apply optimal control techniques to the problem ofmaximizing U0 subject to (10), taking as given p0, pC_t, q_1. For simplicity,we set = = = 0 for all t 0. The result of this straightforwardcontrol problem will be an infinite sequence m0, m1,. .., denoted hereafter

that maximizes U0. Let us write this optimal choice of monetarypolicy as We have already noted that e0 will in general be a functionof p0, q_1 and the entire sequence The first step of this sequencetsth0.

rGiven th0, e0, p0, and q_1, we can use (10) to find p0, q0. Suppose

now that at time 1 the policymakers reoptimize, in order to maximize U1subject to (10). Once again, a simple control problem will yield a sequence rm1, m2, ..., now denoted as {th}r. In general, will not equal t 1,

so that the government at time 1 will not want to carry on with the optimalplan as of time zero. If the government at time 1 is not bound (e.g. by aconstitution) to carry out t

As Kydland and Prescott stressed, we cannot simply assume away thisproblem by letting the initial government choose th0, the next choose th1,etc.; i.e. by letting each succeeding government optimize anew, using theoptimal control solution (this is close to what Buiter (1983) and Miller andSalmon (1983) propose, incorrectly we believe, as discussed below). Theproblem is much deeper, for the following reason. The choice th0 is optimalonly under the assumption that it is followed by th1, th2 It has noparticular attractiveness given that it will be followed by th1 and otherme # for t 2. Moreover, the exchange rate e0 will be a function not of{th0}, as the original government's solution assumed, but rather of theactual that will be selected.

Phelps and Pollak, and Kydland and Prescott, provided the answer tothis difficulty. Unless the original government can act to bind all futuregovernments, it must optimize with the full knowledge that all future egovernments will be free to optimize. A time consistent equilibrium is onein which each government optimizes its policy choice taking as given thepolicy rules (or specific policy actions) that future governments will use.With a finite time such an optimization is easy to carry out. LetXT represent the inherited state of the economy in the final period T. Inour example XT would be the vector <PT' qr-i>. Given XT, it is easyto find the best policy mT =fT(xT) that maximizes At time a:

T— 1, the penultimate government knows that its successor will followmr =fT(xT). It is then an easy task to maximize EtTC subject to(10) and the constraint mT =fT(xT). This second optimization will yield athe rule mT_l =fT_l(xt_l). By backward recursion, every governmentcould thereby find a policy rule that is optimal given the rule that t

a


of succeeding administrations will follow. Such rules will be credible to the

ity, private sector (e.g. the asset holders in the foreign exchange market)ard because each government is doing the best that it can given the freedomrter of action of future governments.ary in an infinite-horizon setting, the solution of the time-consistency issueion is a bit more complex, as we shall soon see. The problem is that there isnce likely to be a multiplicity, perhaps an infinity, of policy rules that have the

• property that they are optimal given that future governments will also• ose choose the rule. There is an embarrassing abundance of time-consistent

U, policies. Not only is it hard to find all of these solutions, but it is notnce necessarily straightforward to choose among them.

1, In summary, there are typically two types of equilibria in multiperiodnal planning problems. The first type assumes that the initialgovernment can

• a pre-commit to an entire sequence of moves, or to a policy rule. For thistype of problem, optimal control suffices. The second types of problem

this more realistically assumes that each government can make its 'move,' but• the, cannot bind the hand of future governments. It must therefore optimize,

the taking as given the freedom of choice of future governments. Beforemd proceeding to the multicountry setting, it is useful to study some more[he technical aspects of these two approaches.iialno Pre-commitment equilibria

her There are two types of pre-commitment equilibria. In the first, theof government selects an entire sequence that by assumption will be

the carried out at all future dates. In the second, the initial government selectsa = .

. .) that is also assumed to bind all future governments.to The first equilibrium is termed an open-loop solution, and the second, a

ure closed-loop solution. Both solutions will tend to be time-inconsistent,ure except in special cases, in the sense that future governments will want to

deviate from the original sequence (in the open-loop case), or the originalthe rule (in the closed-loop case), even if they believe that other governmentsse will abide by the original plans.

We now calculate the optimal open-loop equilibrium in order toIn pinpoint the source of the time inconsistency. Starting with (10), we write

ISy the elements of the A matrix as the B matrix as and the C matrixme as cjj (the specific values of and are given in the footnote •

OW preceding equation (10)). In fact C can be ignored under our simplifyingto assumption = = = 0 fort ) 0. ThuSpg÷1 = a11pg + a12 +

a13 + a14 + while similar expressions hold for p1's and et+i.Int The goal is to choose the sequence that maximizes U0 in (11) subject

to(l0). To solve this problem, we write down the Lagrangian 2' as follows:

a

282 Glues Oudiz and Jeffrey Sachs

max 2 —(i)

+/L1, + a12 + a13 + a14 e1 + b11 mt

+ + a23 + a24 eg + b21mt —pg]

+1u3 t÷i[a3jpg + a32 + a33 qt—i + a34 + —

+1u4 t+i[agipg + + a43 qg_1 + a44 eg + b41 mt —(12)

As is well known, and are shadow values which describehow U0 is affected by different inherited values of Po' and q_1. Inparticular, /L1,0 = 3U0/ap0; it2,0 = and ,Lt3,0 =

By analogy, equals that is it4,O measures the change inintertemporal utility for a small change in e0. Unlike p0, and q_1,however, the policymaker does not inherit e0, but rather determines e0 asa function of the policies that are selected. Because e0 is a policy choice,a necessary condition of the optimization must therefore be thataU0/ae0 = = 0. At the optimum, it41 will equal zero at t = 0.

The time consistency problem arises because along the optimumsequence will (in general) not always equal zero. (6u4g will followa difference equation of the form described in the Appendix). Since willtend to move away from zero, reoptimization at any twhen + 0 wouldlead to a new sequence of m such that would again start at zero (anecessary condition of the optimization). From a technical point of view,the open-loop sequence is time consistent if and only if the equation for

can be satisfied with g 0 for all t 0. If this condition is met, thenfuture governments will choose at all dates t even if they are notbound by the original government. If the condition is not satisfied, theopen-loop solution makes sense only if future governments are not allowedto reoptimize.

Consider a simple illustration using our model. We select simulationvalues for the key parameters of the model, as shown in Table 7.1. Theeconomy inherits a ten-percent domestic inflation rate, and lagged fullemployment (i.e. p0 0.10; p1 0.0; = 0.0; q_1 0.0). With aconstant exchange rate (e0 = 0), CPI inflation will equal ten percent in turperiod zero (i.e. ir0 = 0. 10), while a currency appreciation can reduce theinitial CPI inflation rate. Given our parameter values, the optimalsequence is sharply contractionary at t = 0, so that output is pushedbelow zero, with the goal of reducing inflation. The real exchange rate

+ —p0 appreciates at t = 0, 4.7 percent above its long run value, with {t

the currency appreciation helping to export inflation abroad. Figure 7.1 in

shows the optimal paths of inflation, output, and the real exchange rate. in

(1984 is taken as t = 0). is

• in

—1'

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Table 7.1. Parameter values

1.00 e=0.50fl=0.75 0=0.30y=O.OO A=0.756'=I.SO o=I.50

,,//,/

11

.2)0.10-

be008

0.06 -

0.04

0.02

0.00 -

—0.02 -

—0.06

—0.08 I I

1980 1990 2000 2010 2020 2030

OutputInflationReal exchange rate

7.1 Open-loop control in the one-country model

Consider the behavior of as shown in Figure 7.2. After t 0,turns positive, meaning that an increase in e would raise welfare. Fromthe point of view of the government at time t = 3 (1987), for example, theoriginal plan is too contractionary, since a currency depreciation wouldraise welfare. A new optimization at t = 3 would lead to a new sequence

with rn3 > ph3. This is shown in Figure 7.3, where we superimposeand Loosely speaking, the initial government, at t 0, has an

incentive to announce a stern set of future monetary policies in order toinduce a currency appreciation at t = 0, and thereby to reduce (whichis otherwise very high). Of course, e0 can be reduced by extremely low m0

a

284 GlUes Oudiz and Jeffrey Sachs

0.025

0.020

0.0 15

0.0 10

0.005

0.000

7.2 Shadow price on the(one-country model)

exchange rate in open-loop control

and higher for t 1, or by more moderate m0 and somewhat lowerfort 1. The optimal policy is to opt for moderate m0 and low future m,rather than extremely restrictive m0, since the approach with restrictivefuture m achieves the same currency appreciation with a somewhat lowerloss of initial output, q0.

Thus, from the perspective at t 0, it is worthwhile to commit futurem to low values for the sake of e0. However, from the perspective of futuregovernments, e0 is a bygone, and m should reflect tradeoffs in the presentand future, not the past. Thus, by the time a future government assumesoffice, part of the original incentive to keep m low has disappeared,and the new optimization in period t consequently yields a higher valueof mg.

It is interesting to note that there is a single special case in which theopen-loop policy is also time consistent, and that is when = 0 in theoriginal model (i.e. output is not affected by the real interest rate). In thatcase, u4e 0 satisfies the equation for P4t derived in the Appendix.3 Froman economic point of view, wheno = 0, only the exchange rate e0, but notthe sequence of future m, affects q0 and ir0, so that there is no reason toprefer one path of m over another as long as they both lead to the same

1980 1990 2000 2010 2020 2030

e0.

g(

cc.

avi

0.0

—0.1 —

1980 1990 2000 2010 2020 2030

7.3 Reoptimization of open-loop control in 1987 (comparison withoriginal solution; one-country model)

e0. The same is true about all future ee. This property allows the originalgovernment to specify a path that all future governments will becontent to honor.

The open-loop equilibrium is the best pre-commitment equilibriumavailable. It is sometimes argued, however, that while governments cannotcredib)y pre-commit future governments to a sequence of policy moves,they may be able to pre-commit governments to a specific policy rule formg. Such a closed-loop rule might not be as good as the open-loop result,but it might be better than no rule at all. There is some merit to thisargument, as we shall soon see. The rule can of course be of varyingcomplexity. We illustrate this case by choosing a simple rule, which links

to the current state of the economy, as described by the vectorSuch a rule is termed memoryless, in that the past history

of the economy, in arriving at <ps, is not permitted to affectm1. We simplify further by specifying mt as a linear function ofand qg_1:

mt (13)


0.8 -

0.7 -

0.6 -

0.5 -

0.4

0.3-

0.2

0.1

M: Open-loop policy with reoptimization in 1987

:rol ill: Open-loop policy

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Our method of solution is straightforward. A solution of the form (13) se

is guessed. Using (10) and the assumption that e0 places the economy on tathe stable manifold, we find U0 as a function of the rule. Implicitly thenU0 = U0(/30, /32, /33). Usinga standard numerical optimization technique,we then proceed to maximize U0 with respect to fin, /3j, /32, to arrive atthe optimal rule Given our assumedparameter values for the structural model, we find:

mg (14)

Note that this is the optimal linear rule for a given x0 hi

q_1> = <0.1, 0.0, 0.0>. For a different starting point, we would m

find a different rule.is

Time-consistent equilibriaThe previous equilibria depend on the unsatisfactory assumption

that future governments can be bound by rules made at an earlier date. inj

Some writers have suggested that macroeconomic policies must thereforebe formulated as constitutional rules, in order to bind successfully at a laterdate. For many reasons, including conflicting views about the correct rules,unwillingness to tamper with a constitution, and the realization that evenconstitutions can be amended at a later date, there is little likelihood that (.1

the macroeconomic policy will soon be etched in constitutional stone. In SI:

practice, therefore, governments must operate with the knowledge thatfuture governments have freedom to change course and will have incentivesto do so, relative to the open-loop or closed-loop optimum, even when thefuture governments share the goals of the earlier governments.

In this circumstance, we can reformulate the policy problem as a gameamong an infinite number of players (i.e. governments), who are identified ii4

by the time period in which they act. The initial move is made by thegovernment at t = 0 (hereafter G0), then by G1, and so on. The payofffunctions for is and the move is

Now, we can think of various types of Nash equilibria among thesegovernments. In analogy to the pre-commitment case, we can think of fn

Nash equilibria in which each government takes as given the moves of other d9

governments, or Nash equilibria in which each government takes as givenpolicy rules of other governments. A Nash equilibrium in moves will becalled 'open-loop,' and a Nash equilibrium in strategies or policy rules willbe called 'closed-loop.'

Consider first the case of open-loop Nash equilibrium. Let denotethe sequence of moves before and after, but not including, period t:in0, m1 An open-loop Nash equilibrium is a


3sequence with the property that for all governments mN is optimaltaking as given

is an open-loop Nash equilibrium if and only if for

all t, mf" maximizes E fi' subject to (10) and given {mN}_,.

(15)

In performing the optimization at period t, the government assumes that4) adjusts to keep the economy on the stable manifold, given the past— history of m, the current policy choice and the assumed future path

N NId t+1' t+2' . .

With this definition, the problem with the precommitment equihbnumis that the resulting path is not a Nash equilibrium among the infinitesequence of governments (this was verified in Figure 7.3). Taking as given

)fl that other governments will play (the open-loop sequence), only the

•einitial government will want its part of the sequence (i.e. rh0). For all othergovernments (in general), there will exist a superior choice of policy.

er Now, consider the 'closed-loop' version of Nash equilibrium, in whichwe assume that plays a rule (or strategy) J, which maps (xe, ...)to rather than just a move As before, define the sequence as

at (f0'f1' ... ...). Now, we define a Nash equilibrium in this strategyspace as follows:

at is a closed-loop Nash equilibrium if and only if for alles

t, ...) maximizes subject to (10),

and given (16)

In general, there will be many such Nash equilibria, some of which (as weshall see) are not very desirable.

if As is typical in such circumstances, we further refine the nature of theequilibrium to include only Nash perfect equilibria. A strategy sequence

is said to be a perfect equilibrium if for any history of the economyfrom time 0 to t (even histories not resulting from a Nash equilibriumduring periods 0 to t), strategies constitute a Nash equilibrium in thesub-game from t to CX). We now define time consistency:

time consistent if and only if is a Nash11 perfect equilibrium. (17)

:e In general, open-loop Nash equilibria, as in (15), will not be perfectequilibria. Suppose, for example, that the sequence th1, th2, ... has the Nash

a property. In most models, including those in our paper, the sequence

288 Giftes Oudiz and Jeffrey Sachs

rn2, rn3, ... will not be subgame Nash (starting at period 2), if m1 is setdifferently from Thus, from this point on, we restrict our search fortime-consistent equilibria to closed-loop Nash equilibria, in whichgovernments take as given the policy rules of other governments.

Unfortunately, even the perfectness concept does not eliminate theproblem of a multiplicity of equilibria. There will in general be many trulytime-consistent equilibria. To narrow the search, we begin with the Insimplest case, in which m1 is a function of the current state (b>(= q1_1>) alone (see Maskin and Tirole (1983) for some justifi-cation for restricting our search to such 'memoryless' strategies). Thus, we theare searching for a function m1 =J(x1) such that:

1

e0:m1 =1(x1) maximizes /.?' subject to (10) and to the

1—11

restriction that m1 = 1(x1) for all I + t. (18) equ

(Note that in this case the government at time I does not actually care aboutthe rules up to time t, since the past is fully summarized in xg). Implicitthroughout is the assumption that e1 is always such as to keep the economyon the stable manifold. In practice, this means that along withf there isanother function h linking e1 and x1: e1 h(x1).

Our strategy is to search for f among the class of linear functions.Although we cannot prove that the resulting function is the uniquememoryless, time-consistent equilibrium, we suspect that it is in fact d[-unique, in view of the linear-quadratic structure of the underlying problem.Consider the necessary conditions for a time-consistent optimum. Letm1 = v0 + ViPt + + y3 q1_1 be a candidate solution (call it the y-rule).Plugging this rule into (10), we can also determine a unique linear rulee1 = that keeps the economy on the stablemanifold. Now, suppose that these rules hold for all t 1. It is possible tocalculate fit U1 as a function of the rule and the state of the economyat t = 1, i.e. x1. Let us call the value of the utility function V?'(x1), where milVY denotes the dependence of utility on the rule v•

At time zero, the 0th government wants to maximize U1, whichequals U0 + fi I'7(x1) under the assumption that future governments will usethe y-rule. Note that x1 = <p1, q0>. Specifically, the initial governmentsolves the following:

max U0+PVT(p1, pg, q0)mo

Subject to:

• (a) e1 = h0 + h1p1 + h2pg + h3 q0 m0

(b) Pi = +a13q_1+a14e0+b11m0


et (c) pg = +a23q_1+a24e0+b21m0(d) q0 = a31 p0 + a32 + a33 q_1 + a34 e0 + b31m0

(e) e1 = +a43q_1+a44e0+b41m0

(0(g) Po' q_1 and given (19)

.1y

he In this optimization problem, (a) is determined by the candidate y-rule.•

(b)—(e) are the structural dynamic equations summarized in (10). (f) is thefi instantaneous utility function (note that ir0 = pg Finally, (g) defineswe the state of the economy for the initial government.

The optimization is straightforward. Using (a) and (e) we can writee0 = (l/a44)[h0 +h1p1 +h2pg + h3q0 —a41p0 — a43 q_1 — b41 m0]. Now

• using (b), (c) and (d) together with the new equation for e0, we have four

• 8) equations that make e0, pg. q0, and Pi linear functions of m0 and the pre-determined variables p0, q_1. Let us write this system as:

• cit e0 =

ny pg =(20)

is q0 =

•

'1 =

ue Now simply impose the first-order condition thatict d[ — + +13 pg, q0)]/dm0 equals zero. By direct substitution wenfl. have:

0 =— + d22p?1 + d23 q_1 + d24 m0

+fi(av'/ap1)d44

t +fi(0J'7/apg) d24+/J(3V'i'/3q0)d34 (21)

re This gives us a linear rule for m0 as a function of p0, q_1 and implicitly

-h(through Vr) the y rule:

m0 = [1 /(d34 + [(d34 d31 + çbd24 d21)p0

nt +(d34 d32 d22)

+ (a d44 + (a d24

+ (a d34] (22)

Under our assumptions, the partial derivatives of V?' are linear functionsof p0, and q_1 (though not easy to write down analytically!). Thus,m0 is a linear rule in p0, and q_1:

m0 = (23)


As long as (23) is the same as the y rule, we have found a stationary,time-consistent rule. That is, for = = Yi' 82 = Y2' 83 = y3, the yrule is validated as a time-consistent policy. Starting at any period sandany state 1, the :th government will choose the y rule given that all futuregovernments will make that choice.

In general, the time-consistent rule must be found numerically (seeCohen and Michel (1984) for an elegant treatment of the one-dimensionalcase for the state vector x, for which an analytical solution is found). Todo so, we start with a finite-period problem, in which = It isthen easy to find the optimal final period rule mT =fT(xT). GivenfT,fT.1is readily found by the type of backward recursion just described. For eachT, we can readily computef0(x0). Denote this rule asfoT(x0) to denote thedependence of the rule on the periods remaining. Then it is a simple matterto find the limiting value as T-+ cc. The rule 1(x0) = lim

can then be verified directly to have the time-consistency, Nash equilibriumproperty for the infinite-horizon game. We provide details of this methodin the Appendix.

Using the parameter values described earlier, the time-consistent rule iscalculated to be:

= — 0.032 + 1.032 + 0.275 qg_1 (24)

As is shown in the Appendix, the open-loop optimal policy can be writtenas a linear function of the state variables and

= qg_1+O.389 (25)

Starting, as before, with 10 percent inflation, we can compute the path ofoutput and inflation for the time-consistent policy, for comparison withthe open-loop pre-commitment equilibrium. In Figure 7.4a, we comparethe inflation performance in the two cases; in Figure 7.4b, we compare theexchange rates; and in Figure 7.4c, we compare the output paths. We havealready seen that the open-loop control holds future governments to anover-coniractionary policy relative to the one that they would select uponreoptimization. Since the time-consistent policy explicitly allows for(expansionary) reoptimization in the future, it is not surprising that the areal exchange rate is less appreciated in the time-consistent (TC) case than Miin the open-loop (OL) case. Simply, agents recognize that futuregovernments will select more expansionary m, and is an increasingfunction of the entire sequence of m. Thus, via the exchangerate effect. In general, in the early periods, as governments in is

the OL case pursue a steady, contractionary policy. After a certain period(shown as tin Figure 7.4c), the inequality is reversed. Both policies reducethe inherited inflation to zero in the long run. Su

Time-consistent rule

Open-loop optimal policy

7.4 A comparison of open-loop and time-consistent policies (one-countrymodel)

Before turning to a welfare ranking of the various policies, we must notea key feature of the disinflation process (pointed out earlier in Buiter andMiller (1982) and elsewhere). The price equation is:

(Pt+i Pg) = + + O(q5 —

= p5+(l —A)(p'+e5—p5) = —A)r5, wherer5 (=is the real exchange rate. Thus,

(Pt +(1 —A) — r1_1) + i/Iqg + O(qg — (26)

Suppose an economy inherits an inflation rate of =p0—p1, with


.

.y,0.10 0.00

Vod 0.08 0.O1

ire0.06 —0.02

;ee

ial 0.04 0.03

• To• is 0.02 —0.04

'r—i

—

tch:he

0.00 —0.05 i I I

1980 1990 2000 2010 2020 2030 1980 1990 2000 2010 2020 2030

- ter a Inflation rate b Real exchange rate

x0) 0.00

'

—0.081980 1990 2000 2010 2020 2030

c Output

25)

ofithireheive

anon:or:heanirengige

inodice

a - - . 4

292 GiBes Oudiz and Jeffrey Sachs

r_1 = q_1 = 0. By simple forward integration of (26) from t = 0, we havegi

(pt+i—pt)=Ao+(l—A)re+* E (27)i—o

Now, for all of the equilibria so far considered, Pt÷i equals zero in the Tlong run (i.e. inflation is eliminated), returns to zero (i.e. no long-run mchange in competitiveness), and qj returns to zero (i.e. long-run full CC

employment). Thus, taking limits of (27), we find 0 = z10 + or B'

se

E = —40/ifr (28) gi-o

All policies have the same cumulative output loss, no matter what is thetime path of exchange rates, money, etc.! Thus, the welfare issue is always g

one of timing, rather than the overall magnitude of lost output. ft

On purely logical grounds, we can rank the welfare achieved by the three tipolicies so far studied: open-loop control, closed-loop control (with ai

pre-commitment), and time-consistent control. The open-loop control isclearly first best, since both of the other solutions reflect the saftieoptimization, but under additional constraints. The closed-loop, linearfeedback rule also must produce higher utility than the time-consistent prule. Both the linear.rule and time-consistent solution choose as a linearfunction of the linear rule is chosen as the best among this class of IV

functions, so in particular it is better than the time-consistent rule. Thuswe know that U0OL ? In general, the inequalities will be strict,though we have already noted special cases (e.g. = 0) in which all of the 0policies are identical. u

Buiter (1983) has recently proposed an alternative strategy for finding pa time-consistent linear rule, which has also been treated at length by Miller ii

and Salmon (1983); (we describe Buiter's approach at length in theappendix). His reasoning is as follows. Consider the open-loop control g

solution, with shadow prices and on the state variables, andon the exchange rate. At t = 0, the initial government chooses policies sothat 0. For t> 0, we know that will tend to deviate from zero.Each government in period t would like to reset = 0. Buiter proposes,therefore, that a time-consistent solution is found by asswning that 1u4 0for all t, and dropping the open-loop dynamic equation for When thisprocedure is followed, we obtain the following linear rule:

= (29)

There are two counts against this proposed solution. Most important, th

it is simply not time consistent. If all governments for r 1 adopt the Buiter 1;

rule, the government at t = 0 would not choose this rule. By following the


procedures described earlier (for calculating the best rule at t = 0 for aye

given rule at t I) we find that the initial government would choose:

mt (30)

hThe logic underlying the Buiter solution seems problematic as well. The

e merit for a government to choose = 0 comes if the sequence of incorresponding to = 0 will in fact be carried out by future governments.

U But, by construction, each succeeding government alters the chosensequence of m. There is simply no attraction to choosing = 0 if the

'8'government knows that its plans will not be carried forward. The private

/ sector understands this point perfectly, by setting et to correspond to the

heactual sequence of m rather than to the sequence planned by eachgovernment. In a nutshell, Buiter's government is naive in assuming thatfuture governments will carry out its open-loop optimum, at the same timethat the private sector is completely on top of the policy-making process,

ith and knows that future governments will reoptimize.

isme Reputation and time-consistency

In the previous section we simplified our search for a time-consistentpolicy to 'memoryless' rules. Such rules make a function of thecontemporaneous state vector but not of the past history of x and m.

of Many policies in the real world depend on the history of a game as much

tus as the current state. In competitive environments, for example, aggressive

ct, behavior by one player at time t — 1 might bring forward retaliation byhe others at period t, as in 'tit-for-tat' strategies. Game theorists have long

understood that such history-dependent strategies can help competingng players to achieve more efficient outcomes than those obtainable fromler memoryless strategies alone.he It turns out that similar complex strategies can help a sequence of:ol governments to achieve a better equilibrium than the one obtained by the

memoryless rule = Consider a compound rule of the sort:SO (a) Government t chooses its policy according to mg = g(xg), as.0. long as all governmentsj < t have also selected policy this way;eS, (b) If any governmentj < t selects in5 # g(x5), then government t

0 selects = where! is the memoryless, time-consistenthis rule. (31)

Suppose now that the rule g(xg) is better than in the sense that if all- governments t 0 choose they achieve utility Ufl> u(. Also, supposent, that g(xg) itself is not time consistent in the sense of(l9): If all governmentster I 1 are known to choose g(xg), it is not optimal for the government at:he t = 0 to select g(x0).

7)


The surprising result is that while is not time consistent, a becompound strategy like (31)(a)—(b) can be time consistent with the result gothat all governments end up playing leading to higher social welfare. seIn the memoryless time-consistency problem, each government takes asgiven the choice of policy rule followed by future governments. If future sugovernments are going to choose = the current government may hihave no particular incentive to choose g. With a compound rule as in (31),the government at time t knows that it affects the policy rule selected by gofuture governments. It takes as given the two-part decision mechanism au

(a)—(b), but it recognizes that if it is the first government to deviate fromg(xg), it will cause all future governments to choose instead of g(xg).Since (10> U1 by assumption, this deviation from imposes a cost,which deters the government from deviating from

Thus, each government operates under a' threat' that future governmentswill revert to play mt = Gametheorists have long recognized that such a threat mechanism is viable only mtif the reversion is credible. For example, suppose that the rule is kn'let money growth obey the open-loop strategy or else each future gogovernment lets money grow by one million percent.' If every government f]takes it as given that future governments hold this rule, then money growth is

will indeed obey the open-loop strategy (governments would seek to avoid ththe hyperinflation that they fear would otherwise ensue). A true mtintertemporal Nash equilibrium is obtained, iii which the open-loop mi

sequence is carried out by every government. The problem here, of course, in

is that the threat of hyperinflation is not rational. Surely, if any government eq

does violate the open-loop rule, the next government will not exercise thethreat. Knowing this, no government really has an incentive to persist in prthe open-loop path. eq

Game theorists therefore restrict the threats to actions that would indeed as

be carried out if deviations from occur (even if, as in the example,the threats need never actually be carried out). It is here that theassumption of perfection of equilibrium becomes important. In the mhyperinflation example just cited, not all subgames are Nash, and thusthe proposed equilibrium is not perfect. To see this, suppose that G0deviates. Even if G1 assumes that all future governments will play thehyperinflation threat, it is not optimal for government 1 to play the threat,Thus the subgame in which government 0 deviates, and all (1 ? I) letm grow by I million percent per period, is not a Nash equilibrium. G1 cando better unilaterally, taking as given the actions of other

As long as the reversion is to i.e. the threat is to return to thetime-consistent rule, the threat is credible. After all, if a government

a ' . . .


believes that all future governments will play J(xj, it is optimal for thegovernment itself to play Every subgame consisting of the infinitesequence of governments is therefore a Nash equilibrium.

Now we argue that by this mechanism the sequence of governments can

resustain any linear rule = as long as the utility from this rule is

ayhigher than the utility from the memoryless time-consistent rule for any

1)We want to show, therefore, that the following strategy for each

by government constitutes a perfect Nash equilibrium, in which = is

always played.

(a) Each government chooses = as long as all governmentsj < t have also selected this rule;

st, (b) If any government] < z selects a different then all govern-ments £ select (32)

• :itsne Now let us examine the incentive of any government to deviate from

= It knows that all future governments will then play But

is knowing that all future governments will play it is optimal for theire government in question to choose m1 as well, by the definition of

f In other words, if a government is going to deviate, the best deviationith is simply to revert to immediately. Thus, the cost of defecting from)id the me = rule is to revert immediately and permanently to the•ue rule. Since utility is higher under / than f, there is never an

incentive to deviate from 1. The equilibrium is since in any subgame

Se, in which a defection from = has occurred, it will be a Nashequilibrium for all governments to revert

he For the case 0 0.0, we have found a rule = that has thein property that and thus have verified that such reputational

equilibria exist in our model. With 0 = 0, and all other parameter valuesed as in Table 7.1, the time consistent rule is:

me = O.l6SPe+ I.165

he The following rule has higher utility for all:us

G0 = =

The loss functions corresponding to these rules are:

let r 1.726 —1.726]an = 1.726

=

.1

I I

1980 1990 2000 2010 2020

luoLc)(One-country model)

a Note that the y-axis has been adjusted by amultiplicative factor for graphical convenienëe.

7.5 The cost of reversion to time-consistent control

We have not found such an example for 0 > 0.0.In an important sense, then, the time inconsistency problem is exagger-

ated, in that many 'pre-commitment' equilibria can probably be sustainedeven in situations where actions of future governments cannot be bound.The memoryless time-consistent equilibrium is the lower limit of what canbe obtained by a sequence of governments, not the only outcome. Weshould stress, however, that time consistency does impose costs, since thefirst-best, open-loop strategy almost surely cannot be sustained as a perfectequilibrium. The reason is as follows. Suppose that the sequence ofgovernments pursues the open-loop solution under the threat of reversionto if it ever violates the open loop rule. We know that it willfollow the sequence to which corresponds a sequence of states,denoted At each t, we may calculate the utility of continuing with

296 Gilles Oudlz and Jeffrey Sachs

0.10 -

0.08 -

0.06 -

0.04 -

0.02 -

0.00 -

—0.02 -

—0.04

T

-j

t;

2030

Since S'—S' is positive definite, we have for all xg that c

= >0.


Table 7.2. Two-country model

Aggregate demand== —pg)+yqg —pr)]

Money demandmt—pt =

•=

Consumer price index= Apg +(l= Ap'+(l

— Domestic price levelPi=Wt

• pt*=Wt*

Nominal wage change=

• =

Inflation7t

irt• Exchange rate

eg+j =

the open-loop sequence, with the utility of reverting to thetime-consistent equilibrium, The threat of reverting to f willcontinue to work only when However, at some pointthis equality is reversed, and the government at that date actually prefersto revert to the time-consistent equilibrium. Knowing that such a date willbe reached, earlier governments will also know that the open-loop path

ler- cannot be sustained. This phenomenon is shown in Figure 7.5, where attied each t, we graph with the calculated along thend. open-Loop path. As long as is positive, the governmentcan at t does not have an incentive to deviate. At time I (here 1987), theWe government prefers to revert to the time-consistent solution.thefect HI Policy coordination in the two-country model

ion The first part of the paper has dealt with economic policy in a singlewill economy. We now extend the same set of techniques to a two-countrytes, setting: The goal is to compare 'non-cooperative' equilibria (NC), in whichvith each country optimizes while taking as given the policies abroad, with

0 - - —- - -

_______________________________________


'cooperative' equilibria (C), in which binding commitments can be made cobetween the two countries. Formally, we treat the cooperative case as one dein which a single controller chooses the policies of the two countries. As atin the early section, we must treat two separate types of equilibria: (1) the prpre-commitment case, in which the two countries (in NC) or the single orcontroller (in C), can credibly pre-commit to a rule or to an infinitesequence of actions; and (2) the time-consistent case, in which no pre- tocommitment in future periods is possible. We turn first to the pre-commit- itsment case. to

Open-loop control and policy coordination SuThe open-loop case is most easily dealt with (policy coordination sd

in the open-loop case is also discussed in Buiter and Miller (1983) and Sachs(1983)). We first append a symmetric foreign-country model to thehome-country model just discussed. The model is shown in Table 7.2. In Frthe NC solution, each government at t = 0 solves for an optimal sequence coof monetary policies taking as given the sequence selected from abroad. toIn the C solution, a single controller chooses and to maximizea weighted average of intertemporal utilities at home and abroad. In view ofof the symmetry assumed between the countries, will equalas a feature of both solutions, with the adjustment paths at home andabroad identical. The key result is that non-cooperative control leads to ofover-contractionary anti-inflation policies the social optimum.Both countries are made better off by a coordinated policy of less rapid foi

disinflation. wi

In general, the dimensionality of the control problem is too high toanalyze the NC case analytically. An important special case, however,allows us to establish analytically the key features of the NC versus Csolutions. Since the findings are insightful, we begin with that special case.In particular, we first assume that aggregate demand and money demand Nare not interest sensitive (a = e = 0 in the original model). This simplifi- frdcation allows us to determine as a function of the current state vector atogether with and mt, rather than as a forward-looking variabledependent on the entire future sequence of policies. Also, to reduce furtherthe dimensionality, we set 0 = 0, so that wage change depends on the levelof output but not its lagged rate of change. Nc'

Denoting the real exchange rate as = p' + et we can writeP1 =Pt+(l and = (Pt—Pt_i)+(lTherefore, from the wage equation, and the fact that Pt = we have

= + (1 — A) — + Note from this expression that inflationaccelerated when > or > 0. In other words, a real depreciationbetween periods t and 1+ 1 causes inflation to accelerate, basically becausereal import prices rise. Carrying out the same manipulation for the foreign


ie country yields = ir'—(I Note that a real

ne depreciation at home causes inflation to fall abroad, while an appreciationat home causes foreign inflation to rise. Here is the nub of the coordination

he problem: each country may have an incentive to contract the economy in

'le order to appreciate the currency and thereby export inflation abroad at• 'te the expense of the other country. Since the exchange rate effects are bound

to cancel out if each country chooses contractionary policies to appreciate

it- its currency, a coordinated policy can avoid the contractionary policies,to the mutual benefit of both countries.

It only remains to determine rg before solving for the two equilibria.Subtracting the foreign aggregate demand schedule from the home

— on schedule we find:- :hs = z = (1 +y)/28> 0 (33)• he

In From (33), we see that the key to a real appreciation is to be moreice contractionary than one's neighbor. The effort towards contraction leads

•

to the inefficiency of the non-cooperative outcome.

ize In any period, and p' are predetermined variables, so that the choiceof mg and m' fix and qt respectively, in view of the money demandschedules. Thus, we may think of the policy authorities as controlling qt

nd and qt directly, and then use the sequences and to find the pathsto of prices and the policies and m' and p+ctq'.m. We now write the home country's in canonical,id form. At any moment, there are two state variables, and and we

write the dynamic system in terms of these states:to IPe+i1 _12 —11 [Pt 1 + *

J — i oj j —v) j Lxl —y)j qg

se.(34)

Note that qg is the control variable, and is an exogenous forcing variable6- from the point of view of the home country. The objective function is againor a discounted sum of quadratic loss functions in qg and lrg:)Ie

er U0 —(i) (35)iel

Note that = =We set up a Lagrangian 2 and take first-order conditions in the

standard way (note that ,U1g is the co-state variable for

2'

+1u1g[2pg + + a(l —y)(qg —

+/L2t[Pg — y) (qg — (36)

ite

1).

ye

ononisegn

a

300 GAles Oudiz and Jeffrey Sachs

First order conditions are:

= 0

= 0= 0 =

=0= 0 = = —y)(qt—qt)= 0 = = Pt + tx(l — —

= 0=itP2t-iIfl = 0

We now invoke a sleight of hand. The foreign country is carrying out anidentical optimization, which by symmetry must yield qe = Withoutspecifying the foreign country's problem, we simply invoke this symmetrycondition as a property of the equilibrium, in order to simplify the first-order conditions. Note that when = equals Pg, so that =

= Pt Using these facts, we rewrite the first-orderconditions as:

(a) 0

(b) = 0(37)

(c) = 0

(d)

By direct inspection of (37)(b) and (c), we can see that the system willsatisfy jU2t = We now make that substitution and also substitute for

to write a 2 x 2 system in and ng:

— 38j (

As long as fi < [1 — — y) this system has a single root within the unitcircle and a single root outside the unit circle (the condition is sufficient,though not necessary).s Denote the stable root as (the superscript Ndenotes non-cooperative case). Thus, the dynamics of inflation are:

= (39)

Starting from an inherited inflation rate ir0, the two economies convergeto zero inflation, with a mean lag of years. af

Now let us consider the cooperative case. Here, a single controller Tchooses and to maximize an average of utilities in the two countries.Since the countries are identical, we may assume simply that the controllermaximizes domestic utility subject to the constraint that qg = for


0.06

0.04

002

an0.00

• DUt 1984 1985 1986 1987 1988 1989 1990try(-St.

• — 0.00'der

—0.01' a

—0.02'

—0.03'

37) : —0.04'

—0.05

—0.06'will

—0,07-

—0.08'

—0.09

38) —0.10

.nitnt,

Cooperation

Non-cooperation

7.6 A comparison of non-cooperative and cooperative control (simplified39) two-country model)

rge

all t. With this constraint, the inflation equation is =iler The Lagrangian for the single controller problem is therefore:jes.Iler 'max 2 = (40)for t—o

a - . •• 4

a Inflation

b Output

1984 1985 1986 1987 1988 1989 1990


The dynamic equation for the first-order conditions of (40) are:

— ['eu (41Lxe÷t J

— 11 Lxt

Note the relationship between (38) and (41). The cooperative dynamicsare found by setting = 0 in (38). is the parameter which measures howlarge a real appreciation is achieved for a given contraction of q relativeto q*. It thus indicates the importance of the 'beggar-thy-neighbor'phenomenon, which each country (vainly) attempts to keep output lowerat home than abroad in order to export inflation. Since the single controllerrecognizes the futility of each country, in a closed system, trying to exportinflation, the controller simply sets = 0. That is the root of the gain tocooperation.

The matrix in (41) again has a single stable root, this time denotedThe dynamics of inflation are now

(42)

It is a simple matter to prove that > for > 0, so thatcontrol results in slower disinflation than non-cooperative control.7Figure 7.6 illustrates the inflation and output paths of the home economyunder cooperation and non-cooperation. The faster disinflation under NCis clearly brought about by increased unemployment (i.e. reduced output)in the early years of the disinflation process. Remember from our earlierdiscussion that the cumulative output loss is the same for all paths thatasymptotically reduce inflation to zero.

Welfare aspects of cooperationAssuming that governments are pursuing appropriate objectives Si!

(e.g. that they use the 'right' discount rate), it is easy to show that the kr

cooperative path, with less extreme disinflation, dominates the non-cooperative path. A simple argument is as follows (direct computation du

would also make the same point). Define the set of pareto efficient (E) pairsof sequences that have the property that U0 is maximized thd

given and U is maximized given U0. It is well known that the set ofpareto efficient pairs may be found by maximizing wU0+(l —w) U with PO%

respect to and for all weights w€ [0, 11. Every pareto efficient di

sequence pair maximizes some weighted average of U0 and U, and everysequence pair that maximizes wU0 +(1 — w) is pareto efficient. In

The cooperative solution, by construction, gives the sequence pair ret

corresponding to w = 0.5 (i.e. equal weighting of the countries). It is theunique solution to the problem. Since the non-cooperative solution also no

yields a symmetric equilibrium, with U0 = U, it must be that < tr

J

International coordination in dynamic macroeconomics

U) —11,16-

ics)W

ye

)r'• ier

lerDrt

to

• C6

• Cooperation+ Non-cooperation

Note that the welfare scale on the y-axis has been adjusted bya multiplicative factor for graphical convenience.

7.7 The gains from cooperation with myopic governments

since otherwise the non-cooperative solution would pareto dominate aknown pareto efficient solution.

We mentioned in the introduction that some critics of cooperation aredubious of the assumption that governments maximize the proper socialwelfare function. In particular, plausible arguments have been made thatthe government's discount rate fiG is less than the 'true' fi. If so,cooperation might exacerbate rather than meliorate social welfare. Thepoint is that cooperation allows governments to pursue a more 'leisurely'disinflation. However, short-sighted governments might already be post-poning the necessary disinflation, in return for short-run gains to output.In an already distorted policy environment, cooperation might furtherretard the necessary adjustment.

To examine this view, we computed the open-loop cooperative andnon-cooperative intertemporal utilities for a range of fl°, holding fixed the'true' fi at.(l.l)' (we use the simplified version of the two-country model

a ' . . - a

—11.14'

303

—11.24 -

—11.26-

—11.28 —0.80 0.85 0.90 0.95 1.00

[13G < fi = (1.1)_Il

• 12)

ive)l.'

- ny4CUt)

ieriat

'Cs

•he

Dfl

irsedof•th

:ry

iirheso

0'

a

304 GlUes Oudiz and Jeffrey Sacbs

for these calculations). For each fiG, we calculate the two equilibria andthen evaluate the social welfare of the resulting paths using fi = (1.1)—'.

As seen from Figure 7.7 non-cooperation dominates cooperation when fiGis sufficiently smaller than fi, and cooperation dominates non-cooperationas long as is 'close enough' or somewhat greater than fi. Of course,for any fiG = fi, open-loop cooperation will necessarily be superior toopen-loop non-cooperation. It is not the level of fi° but the difference ofj3G and fi which might cause cooperation to be welfare reducing.

Policy coordination and time consistencyWe now leave the case of open-loop control and return to the more

realistic assumption that governments cannot bind their successors. In thenon-cooperative setting we are looking for an equilibrium characterizedby rules and =f*(xt) that have the following property: forthe home country,f is optimal at time t given that all future governments I

at home play f and that abroad the contemporaneous and all futuregovernments playf*; while for the foreign countryf* is optimal underthe analogous conditions. Note that is the state vector includingpredetermined variables of both the home and foreign economy. In

t• 1 —/ * C C* *par icu ar, Xt — \Pt, Pt Pg—i' Pt—i, qt—i,There are two key differences with the open-loop model previously

described. First, of course, is the inability of G0 and to bind the entiresequence of future moves. Second is the assumption that each governmenttakes as given the foreign rule rather than the foreign actions, so thatoptimal moves today take into account the effects of today's actions ontomorrow's state vector, and thus on the foreign governments' moves. Itwould be possible instead to calculate a time-consistent multicountry towaequilibrium in which each government takes as given the sequence of future to thmoves (i.e. open-loop time consistency), but we have not pursued that Si1

choice here. open

As in the one-country case, the time-consistent equilibrium is solved as as itthe limit of a backward recursion. (For the calculations that follow, we one-o

revert to the complete two-country model, with non-zero values of sequd

and 0.) Using the parameter values of the one-country model, we arrive unde4

at the following rules: contrj

nit =openthat

+ 0.328 qg+O.O5 qt (43)

Figure 7.8 compares the paths of the home economy output for the now

non-cooperative open-loop and non-cooperative time-consistent equilibria, as wej

As in the one-country model, output losses are smaller in the early periods to a s4

for TC than OL. The inability to bind one's successors causes a bias that

• fore

rized

• forentsIture

• nderding'. In

• uslyntire Time-consistent rule

nent Open-loop optimal policy

that 7.8 A comparison of non-cooperative control: open-loop versus time-s on consistent solutions (two-country model)

Ittowards more expansionary policies and thus more rapid inflation, relative

.ture to the open-loop solution.that Significantly, it is no longer possible to rank social welfare under

open-loop versus time-consistent policies (for non-cooperative equilibria),d as as it was in the one-country model. Remember the argument in the

we one-country context. Open-loop control, by definition, picks the optimal€, sequence; time-consistent policy, on the other hand, reflects an optimization

under additional constraints and therefore is inferior to the open-loopcontrol. In the two-country setting, the same logic does not apply. Theopen-loop sequence is no longer the optimal sequence. Indeed we have seenthat open-loop, non-cooperative control is typically Pareto inefficient.

(43) There is no presumption that adding constraints to the optimization willthe now lower welfare, particularly since constraints are being added abroad

)ria. as well as at home. It is true that the home country can no longer pre-commitiods to a sequence of moves, but now neither can the foreign country. It is truebias that the home country prefers an open-loop to time consistent policy

r toe of


0.00

—0.01

—0.02

—0.03

—0.04

—0.05

—0.06 -

—0.07 -

—0.08 - —1980

I I I

1990 2000 2010 2020


assuming that the other country isfixedat one or the other. With the othercountry's policy fixed, an open-loop policy at home can exactly replicatethe time-consistent sequence, and presumably it can do better.

There are good economic reasons to believe that the time-consistentpolicy may actually dominate the open-loop solution in the non-cooperativegame. The open-loop policy, we know, is over-contractionary relative tothe efficient equilibrium. Moving from open-loop control to time consist-ency causes policy to become less contractionary and therefore pushes theeconomy towards the efficient equilibrium.

Now, let us consider the time-consistent cooperative equilibrium. Herewe imagine that a single controller each period sets m and m*, but nowsubject to the time-consistency constraint. The single cooperative controllermust optimize while taking as given the actions of single cooperativecontrollers in later periods. We should like to determine whether time-consistent cooperation is superior to time-consistent non-cooperation. Aswe have noted in several places Rogoff (1983) has devised an ingeniousexample where cooperation reduces welfare. Simply, time-consistencyleads governments to be over-inflationary relative to the open-looppre-commitment equilibrium. Cooperation further exacerbates this over-inflationary bias by removing each government's fear of currencydepreciation.

Interestingly, our results run counter to Rogoff's: cooperation issuperior in welfare terms to non-cooperation: While the cooperativesolution is more inflationary (see Figure 7.9), as we might expect, it is notoverly inflationary in a welfare sense. The less rapid disinflation merelycorrects the contractionary bias of the non-cooperative case. The key pointhere is as follows. In the symmetric country model, the single controlleralways adopts symmetric rules so that et = 0 for all t. Since the exchangerate is the sole potential source of time inconsistency in this model, andsince it is always equal to zero, the cooperative time-consistent solutionis also the open-loop cooperative solution. For a cooperative controller,there is no time-consistency problem in our model (since the countries aresymmetric). The single controller can reach the first-best optimum solutionfor open-loop cooperative control.

In sum, we have shown examples where cooperative control is moreinflationary than open-loop non-cooperative control and time-consistentnon-cooperative control. In both cases, the cooperative solution is welfareimproving relative to the non-cooperative equilibrium. In view of Rogoff'sexample, it will be difficult indeed to set out general principles on the gainsfrom cooperation under the constraint of time consistency. Comparing ourexample with his, the key difference seems to rest on the source of thetime-consistency problem. In Rogoff's case, the problem arises from


:her

tenttiveetoiist-the

lerelow

— Her- tive

me-As

ous0.00—

ver-0.00

isLive

notely)iflttierigetndOfl

er,areon

irersinsurhe

0.10 -

0.08-

0.06 -

0.04 -

0.02-

a Inflation

1990 2000 2010 2020 2030

b Output

—0.01

—0,03

—0.04

—0.05

—0.06

—0.07 -

—0.08 I I I I

1980 1990 2000 2010 2020 2030

Cooperation

Non-cooperation

.7.9 A comparison of non-cooperative and cooperative control: the caseof time-consistency (two-country model)


forward-looking wage setters and cooperation exacerbates the problem. In whour model, the problem arises from forward-looking exchange market anparticipants, and cooperation eliminates the problem. 4

ConclusionsThis study represents work in progress on the gains to coordination

in dynamic macroeconomic models. Our focus has been purely methodo- oflogical, and preparatory to attempts at a quantitative assessment ofinternational policy coordination. The methodological issues arise fromthe wide variety of possible equilibrium concepts in multicountry dynamicgames. The games can be solved under the assumption of pre-commitmentversus time-consistency; open-loop versus closed-loop behavior; andnon-cooperative versus cooperative decision-making. These three dimen-sions are all independent, so any choice along each dimension is possible.

Moreover, in some cases there may be multiple equilibria. For example,there are probably many time-consistent, non-cooperative equilibria thatdepend on the 'threat-reputation' mechanism outlined in the paper. As yet,we have made no systematic attempt to search for such equilibria.

This work should now be used to gain empirical insight into thecooperation issue. For all of the discussion surrounding time consistency,

Iifor example, there is not a single empirical investigation of its importance inin the macroeconomics literature. Similarly, there are no reliable measures anof the gains to cooperation the simpler, pre-commitment equilibria. Such finquantitative work deserves a high priority.

thAppendix I

We shall present in this appendix the derivation of the four policy rules•discussed in this paper. All of these rules are obtained as the stationarylimit of backward recursions using a methodology similar to Basar and (AOlsder (1982) or Kydland (1975). The only significant difference with theseauthors is the fact the followers' actions are represented here by aforwardlooking variable, the exchange rate.

Let us consider a two-country world. The world economy is characterizedby an n-dimensional vector of state variables, xt and the domestic currencyprice of the foreign currency is In each country the authorities seek tomaximize a welfare function I = 1, 2, and can use a set of policyinstruments denoted where is an mi-dimensional vector. The Sti

dynamics of the world economy can be represented by a system ofdifference equations.

(Al) W= + + J to


__

In where denotes the stacked vector of instruments for the world economy

ket and A, B, C, D, F and G are matrixes of parameters. Note that matrixesA, B, C are defined differently than matrixes A, B, C in the rest of thechapter.

Let us denote by the vectors of targets for each country. and T2t

•

are linear functions of the state variables, the exchange rate and the valuesio- of the policy instruments:

•

= Xt + + I = 1,2 (A2)

nic where M2, and are matrixes of parameters.ent The optimization problem for the authorities of country I is then:iriden- Max — E1e

t—o

• )le st. (Al), (A2)

hat where is a matrix of parameters and is a discount factor.jet,

The time-consistent solutionthe The time-consistent, non-cooperative equilibrium is found as a

limit to a finite-time (Tperiod) problem, for Tiarge. The solution is derived• tice in two steps. The finite-horizon problem is solved for the last period, T,

res and then it is solved for period t given the solution for period t + 1. Weich find the limit of the rule for period 0, as T-÷ co.

In period T+ 1 we assume that the exchange rate has stabilized, and thatthe value functions V1T+l(xT+l) and are equal to zero:

= eT (A3)les = 0 = 1, 2 (A4)trynd (Al) and (A3) imply that:

eT (1 —F)'(DxT+GUT) = JTXT±KT UT (A5)

Given XT, the authorities of country 1 choose such as to maximizeed welfare in the last period:•cy

to given XTcyhe Substituting (A5) into (A2) leads to the following first order condition:of —F)' Gj'Q,[N1+L1(l F)'G} UT

= + L1(l — F)' G1] Q,[M, + L1(l — F)' D] XT (A6)

1) Where and G2 are the submatrixes of N, and G which correspondto

4

310 Gilles Oudiz and Jeffrey Sacks

A similar condition can be derived for country 2 which yields: wh

MMTUT=-NNTxT U1

where MMT is an (m1 + m2) x (m1 + m2) dimensional matrix. We shall givebelow the explicit forms of MMT and NNT as functions of and KT.Thus under the assumption that MMT is invertible we get a linear policyrule: (en

uT=rTXT (A7)

Substituting (Al) into (A5) yields:U21

eT = (1 —F)-'(D+GrT)XT = HTxT (A8)

Finally we can derive the value functions V2T(xT) for period T:Th

1=1,2M

where

i= 1,2(A9)

us now consider period t assuming that r1÷1, and are given,

= (AlO)

= (All)

= (A12)

Substituting (A12) into (Al) yields:

et=Jgxg+KtUt (A13)

where

f = Iset

(A14)

Jto1coq

The value function of country 1 for period t is defined by: no4

= given Xe (A15)di

Substituting (A 13) into (Al) and (A2) leads to the following first orderconditions:

[(N11 + L1 K11)' Q1(N1 + L1 K1) +/31(C1 + BK11)' S1141(C+ BK1)] U1 thi= — [(N11 + L1 K11)' Q1(iil1 + L1 pr

+fl1(C1 + BK11)' + BJ1JJ (A 16) difop

International coordination In dynamic macroeconomics 311

where and C1 are the submatnxes of and C corresponding toU1g.

A similar set of conditions holds for country 2. We thus obtain:

= — xg (Al 7)

where is an (ni1 +m2) x (m1 +m2) dimensional matrix and NNg is an+ m2) x n dimensional matrix.

k7) Let us divide and

N A= I. —

( 18)

Then we have:

= + Ku)' Qg(Ngj + Lg Kjg) + +(A19)

2 = + Qg(Mj + +/ij(Cg + BKgg)' +(A20)

'en, These formulae hold for period T with and KT defined as above and

10) SgT+1 = 0. Finally we can derive I, and

11) (A21)

12) (A22)

13)Sgg+1(A i = 1,2 (A23)

We have thus obtained both recursion rules and starting values for theset of matrixes I, Hg, and We define as the time consistent solution

14) the stationary solution to which this system converges for t = 0 as T goesto infinity. We do not know of any general result concerning theconvergence of this process. However in our empirical applications we havenot run into major problems. Cohen and Michel (1984) show that in a one

15)dimensional case this kind of a recursion does have a fix-point.

ler The open-loop solutionThe open-loop solution corresponds to a one-shot game where

the authorities announce at time zero the whole path of their policies. Itthus does not by definition require the use of a backward recursionprocedure. The set of dynamic equations formed by the state variable

6) difference equations and the first-order conditions corresponding to theoptimal control problem of the authorities could for example be solved

a

a


explicitly by using the method proposed in Blanchard and Kahn (1980)or numerically with a multiple shooting algorithm (see Lipton, Poterba,Sachs and Summers (1982)). However, we shall present here a backwardrecursion procedure which leads to a simple algorithm.

The optimal control problem faced by the authorities of country i leads Eç

to the definition of the Hamiltonian

= + + ++ + — (A24)

where is the vector of co-state variables or shadow costs which the WI

authorities of country I associate with each of the state variables and, rer

similarly ,Uje÷j is the co-state variable corresponding to the exchange rate.8The set of first-order conditions is then:

= = 0 (A25)

= Q1 + +flj = (A26)

= =

Let us first of all derive the recursion equations at period t. One major wli

difference with the time consistent case is the existence of /tg, the co-state ani

variable corresponding to the exchange rate at time t. Since e0 is notpre-determined, it can be set freely by the authorities in the initial periodby announcing a proper path of future policies. Its shadow cost in the firstperiod, is zero. is thus a predetermined variable equal to zero in thefirst period and has to be added to the vector of state variables, whenthe recursion relations are defined.

More precisely we shall assume that the problem is solved for t + I andthat the following relations hold:

eg+i = xt+i + ,ag+i (A28)

(A29)

= f'e+iXt+i+Yt+t/Lt+i (A30)

Let us now define the following matrixes:

At2 1=1,2 (A3l)

Al3 Al3 A13=flfF wh

Akj[4L

2] (A32)

-k


80) A11 A12 A,3]ba, A1 A21 A,, A2, (A33)ard A3, A,, A,,]

adsEquations (A25) to (A27) can be rewritten in matrix form:

A1 = 0 0 '2n + 0 0[xe]

(A34)

24) Ite+, 0 0 0 p, ] o '2

where '2fl and '2 denote identity matrixes of dimensions 2n and 2the respectively. Then using equations (A28) to (A30) we get:nd,

— te.8 eg = (A35)

Tt = (A36)25) r, 1

26) Pe+i = /Lt÷, (A37)

27)/Lt+, LPt j

where and are the stacked vectors of targets and co-state variables-. and

ate B,, 0 0riot = 0iod

0 T 0irst 2

theien A,, = 0

0 0nd

B — [Mi+L,J,]• B — IN,+L1K,1 B — [L,R,

'8

" [M2+L,J,J " [N2+L,K,j ' 3t

=

(F— B)—'

fuel= (A38)

1) Li's iwhere:

[0 0 0 0 02) MM, = A, A,,

—0 0 I,,, ; NN, = A, A,, — 0 0

[0 0 0 0 1,

a

314 Gilies Oudiz and Jeffrey Sacks

From (A38) we can derive 1, and where the two last wivariables are defined by:

Lastly we get: Eq

= Jt+KJ't+Re11t=

We now need to obtain starting values for the recursions thus defined.If we assume as above that the exchange rate stabilizes at time T and that

Th

RrO =0;

The open-loop solution is the stationary limit to which this recursionconverges. It should be noted that here the policy rule is not only a functionof the state variables, but also of the costate variables

Let us give a simple example in the case where each country has a singlepolicy instrument. The policy rule is = + where y is a (2 x 2)matrix. We also have:

an=

which, given the policy rule, yields:

=wi

Thus we finally obtain

=

The policy rule appears to be of a more complicated form than the timeconsistent rule. It is a function not only of the current state variables butof the lagged values of these state variables and of the lagged moves.

The Bwter solutionBuiter (1983) proposes a solution to the time inconsistency T

problem which we discuss in the paper. Formally his strategy amounts tosetting equal to zero and suppressing equations (A37).

Using the same notation the set of first order conditions becomes:

A1 1 =Pt co


where1St A

AL"21 "22

Equations (A28) to (A30) become:

= (A28')

Pt+i = (A29')

= (30')iat

Then we get:

eg=Jgxg+KgUe=Htxt (A35')

= + (A36')

.on [Tt1 = A22 [ +A32x2 (A37')

LPt+iJ LPe Jwhere

gleA

[B22 01. A — 1B12oj ' — LzIe÷i(A+BJt)

and finally:

=

where

MM2 = A1 A22—

neNN2=A1A32

Ut From (A38) we derive 1 and A2 which give H2:

H2 = J1+K21

The system of recursive equations thus obtained is solved backward fromcy T with the same starting values as above:to

= (1 —F)1D; KT(l —F)'G; 4T4-1 = 0

The optimal linear ruleThe problem here is to derive the optimal linear rule, i.e. the

constant feedback rule which yields the highest welfare for the authorities

a

316 Gilles Oudkz and Jeffrey Sachs

of each country. It can be divided into two steps. The first step consistsin obtaining for a given rule

r= F 'lsuchthatL'JU, = the value of the welfare foreachcountry, W,(fl and W,(fl. Then, 2

in a second step, the optimal values of and 1 are calculated using anumerical gradient method. We shall not discuss here the second step. Thefirst step is again solved by backward recursion which proved moretractable for the repeated calculations imposed by the gradient method.

Substituting = into (Al) yields:

Ixt+il — [A+Cf' B1 Ixti (A39)— LD+Gr F] Let]

For period T assuming eT+l = eT yields

eT = (1—F)-' (D+GflXT = HTxT (A40)

Then if we assume: et+i =

= (F— B)1 + CT')—(D+ Gfl] xg (A41)

the recursion is thus simply

= (F— B)-' + Cfl—(D + Gfl] (A42)

which, starting with HT, has a stationary solution for values of the 3

parameters such that the transition matrix in (A39) has only one eigenvaluegreater than unity. More precisely:

urn H0=

where C22 and C2, are submatrixes of C, the matrix of row eigenvectorsof the transition matrix defined by

C1, C,2nxn nxi

C =lxn lxi

NOTES

* We thank Warwick McKibbin for very able research assistance. We also wish 5to thank Dilip Abreu, Antoine d'Autume, Daniel Cohen, Jerry Green, andBarry Nalebuff for helpful comments. We also thank DRI for generously


allowing us the use of computer facilities. This work was begun while SachsIStS was a visiting professor at the École des Hautes Etudes en Sciences Sociales

in 1983, and was completed while Oudiz was a visiting scholar at MIT andthe NBER during 1984. Oudiz gratefully acknowledges the financial supportof NATO and the Fuibright Fellowship Program.

I See, for example, Nordhaus (1975).2

ien,—l—(*+0)cr4

Arhe A = —(6+crp—crA)4 —cr4 —cr04loreod.

1—A+(ifr+0)4[8+cy(1—A)]I—A

[8+ cr(l — A)] A1 —A)]

crp(ifr + 0)4

op4—p

F 00 00 y4

L(8+cr(l —1 )'/LLI

k42)where 4 = [1 +cr(u—%lr—0)]'

the 3 Using the notation of the appendix, it is readily checked that if r = 0, C and

slüe N1 in (Al) are null matrixes, and G in (Al) is equal to —p. This implies thatthe money stock has no direct effect on either the state variables or on thegovernment's targets: output and inflation. Thus the first-order condition(A25) reduces to = 0.

4 This point is easily proved by considering the following change of variables:

=where =

The differential system (38) becomes:

1/fl+ç6%(t2 —I/fl=

cl-,-1 0 0 1/fl

This system is saddle point stable under the conditions discussed in the textand has one stable root Af' and two unstable roots and I/fl. One variablelTr is backward looking and are forward looking. Given that 1/fl> 1,it is clear from the third equation that along the stable path must alwaysbe equal to zero, so that = —/L2t for all t.

vish 5 The toots of the system can be found by solving the characteristic equation:andusly A2.—(&i+ = 0, where = [I


We assume w > 0. To show that there is exactly one stable root 0 < 1 andone unstable root 1 <Ac', observe the values of the characteristic equation C(A)at A = 0 and A = 1. C(0) = (l/fl)w > 0 and C(1) = [1/fl—I] Blaifra(l —y) <0. Also, for A I, C(A) > 0. Thus, there is exactly one rootbetween 0 and 1, and one root exceeding 1. Bw

The stable root is

The unstable root is:

(w+ Cal

6 The roots for the cooperative case can be found by setting a = 0 (i.e. &j = 1) Coin the equations for the roots derived in note 5.

The stable root is

= (I + 1/fl+ — [(1 + 1/fl+ — 4/fl]i. Do

The unstable root is Ky

= + +(

7 It was shown in note 6 that = when a = 0. To prove that > for ¶

a > 0, we need only show that > 0 for all a. We know that= 0. Consider

Li

= M

We want to prove that the last expression is negative. We know < 0.Therefore,

1/fl)+4/fl2+(w+ <(u+M

orOi

(u— <Taking the square root of both sides and dividing gives Ph

(u —1/,8+ {(u — l//3+ < 1

Substituting into the expression for we see

<0 for all a.Thus > 0 for all a.

8 Note that in the paper the notation is slightly different, being the co-statevariable corresponding to the exchange rate in the one-country case.

REFERENCES

d'Autume, A. (1984). 'Closed-Loop Dynamic Games and the Time-inconsistencyCO

Problem.' Brown University, processed, June. 01

Barro, R. J. and D.B. Gordon (1983). 'Rules, Discretion and Reputation in a aiModel of Monetary Policy.' Journal of Monetary Economics 12, pp. 101—21. cc

*


A Basar, T, and U. J. Olsder (1982). Dynamic Noncooperative Game Theory. AcademicPress.

Blanchard, 0. J. and C. Kahn (1980). 'The Solution of Linear Difference Models

root Under Rational Expectations.' Econometrica 48.Buiter, W. H. (1983). 'Optimal and Time-Consistent Policies in Continuous Time

Rational Expectations Models.' NBER Technical Working Paper.and M. H. Miller (1982). 'Real Exchange Rate Overshooting and the Output

Cast of Bringing Down Inflation.' European Economic Review 18, May/June,pp. 85—123.

Calvo, G. A. (1978). 'On the Time Consistency of Optimal Policy in a MonetaryEconomy.' Econometrica 46, no. 6.

= 1) Cohen, D. and P. Michel (1984). 'Toward a Theory of Optimal Precommitment.I. An Analysis of the Time-Consistent Solutions in a Discrete Time Economy.'CEPREMAP.

Dornbusch, R. (1976). 'Expectations and Exchange Rate Dynamics.' Journal ofPolitical Economy 84, pp. 116 1—76.

Kydland, F. (1975). 'Noncooperative and Dominant Player Solutions in DiscreteDynamic Games.' International Economic Review 16, no. 2.

(1977). 'Equilibrium Solutions in Dynamic Dominant-Player Models.' Journalfor of Economic Theory 15, pp. 307—324.

that and E. C. Prescott (1977). 'Rules Rather than Discretion: The Inconsistency ofOptimal Plans.' Journal of Political Economy 85, no. 3.

Lipton, 0., J. M. Poterba, J. Sachs and L. H. Summers (1982). 'Multiple Shootingin Rational Expectations Models.' Econometrica 50, September, 1329—1333.

Maskin, E. and J. Tirole (1982). 'A Theory of Dynamic Oligopoly, Part I.' M.I.T.<0. Department of Economics Working Paper No. 320, November.

Nordhaus, W. (1975). 'The Political Business Cycle.' Review of Economic Studies42, pp. 169—190.

Miller, M. and M. Salmon (1983). 'Dynamic Games and the Time Inconsistencyof Optimal Policies in Open Economies,' mimeo, University of Warwick.

Oudiz, G. and J. Sachs (1984). 'Macroeconomic Policy Coordination Among theIndustrial Economies.' Brookings Papers on Economic Activity.

Phelps, E. S. and R. A. Pollak (1982). 'On Second-best National Savings andGame Equilibrium Growth.' Review of Economic Studies 49, pp. 185—199.

Rogoff, K. (1983). 'Productive and Counter-productive Monetary Policies.'International Finance Discussion Paper 223 (Board of Governors of theFederal Reserve System), December.

Sachs, J. (1983). 'International Policy Coordination in a Dynamic MacroeconomicModel.' NBER Discussion Paper No. 1166, July.

tate

COMMENT JORGE BRAGA DE MACEDO

Between the Fall of 1982 and the early Summer of 1984, Sachs has writtenthree (or four depending on the count) papers on international policy

flCYcoordination. In the Spring of 1983, with the collaboration of Oudiz, lineoutput accelerated tremendously. It also became more quantitative: only

.n a a few months ago, Gilles and Jeff found the gains from international policy-21. coordination to be empirically modest.

a

320 Comment by Jorge Braga de Macedo

Unlike the remarkable Brookings paper, however, this one has no punchline. Called 'research in progress' in the conclusion, it sets out to elinvestigate three points: whether dynamics make a difference; howinternational policy coordination interacts with the political business cycle;and the implications of the inability of governments to bind theirsuccessors. The last point, by investigating time-consistent solutions, is anelaboration of the first. The second point is discussed graphically towardthe end of the paper. It is ignored in the discussion to follow. The samefor the empirical significance of time-consistency, which deserves 'highpriority' in the authors' concluding judgement. Whilst in agreement, I bdeleted remarks on empirical models of policy coordination to share the'pure methodological focus' of the paper.1 The model used to illustrate rethe method will be discussed first. It is presented in Table 7.2 whileparameter values used in the numerical simulations are in Table 7.1.

The first pair of equations defines the IS curves at home and abroad:q=6r+yq*_q(i_p) (1)

Since the parameter y is set to zero in Table 7.1, trade between the twocountries is unresponsive to output (marginal propensities to import arezero). As a consequence, the coefficient on the real exchange rate (r) is tobe interpreted as the average propensity to import times the sum of thetrade elasticities less one times the inverse of the (common) marginalpropensity to save. Unitary elasticities, and an average propensity toimport of 0.25 (the same as 1—A, the share of foreign goods in theconsumer price index) would give 6 = 1.5 — as in Table 7.1 — if the savings fr

propensity is 0.17. But the experiments where 8 is infinite require infiniteelasticities which are implausible in a two-country world. The parametero. represents the real interest semi-elasticity of investment. Given the valueof the savings propensity and a share of investment like the share ofimports, the value u = 1.5 implies an elasticity higher than the steady-statereal interest rate. If there is no steady-state inflation, a nominal interest th1

rate of 10% would given an elasticity of 0.125. Since the nominal interest m4

rate is deflated by the proportional change in domestic prices rather thanin the consumer price index, the change in the real exchange rate does not al

affect investment.2 N

The second pair of equations defines the LM curve at home and abroad. Ic,

2.m—p=csq—ei (2)

Money balances are also deflated by domestic prices, so that there is nodirect effect of the real exchange rate on money demand.3 The parameter al

is set to one in Table 7.1. The value of e = 0.5 implies that the interest is

elasticity, of money demand is one half of the steady-state level of the ch


ich interest rate or 0.05. This makes it 2.5 times smaller than the real interest

to elasticity of investment.Equations (1) and (2) can be combined into an aggregate demand curve

which involves the real exchange rate, the domestic real money stock(denoted by a bar) and the rate of domestic price inflation (denoted by y):

• an(3)

• igh Unlike the Brookings paper, which introduced an elegant portfoliot, 1 balance model, this one makes the usual assumption that domestic andthe foreign money are perfect substitutes (e = is). The rate of change in theate real exchange rate can therefore be expressed as:nile

1• r=_(q_q*_rn_rn*)_y+y* (4)

d:

(I) Substituting for q and q* from (3), there obtains a steady-stateproportional relationship between the real exchange rate and relative real

:wo money balances, with coefficient Using this in (1), we get output as:are

2oH-e(5)

the ' 2(o + c) 2(cr + e)

Suppose the two countries set m and m* so as to minimize a quadratic

thefunction of the deviation of domestic output and the consumer price indexfrom steady-state

ngsiite U = —A)r]2} (6)terlue Then a price rigidity p = = p will imply a loss of ç5p2 if both countries

of set ifi = = 0 since it is evident from (5) that, in that case, q = 0.

ate If, instead, each country attempts to appreciate the exchange rate, taking'est the other country's money stock as given, they will bothend up with a lower'est money stock. Loss in this Cournot-Nash non-cooperative solution isian magnified by ç5[(l — + + €)]2. Other non-cooperative solutions,

allowing for a non-zero conjectural variation, imply a lower loss than theNash.5 Still, with the parameter values of Table 7.1, the squared term is

ad. less than 1 %, so that loss varies from 2p2 in the cooperative solution to2.02 p2 in the Nash non-cooperative solution. This summarizes the results

(2) of a static policy game.no Coming back to the dynamics, they are certainly not confined to (4)ter above. There is also an equation for the change in nominal wages whichest is a function of the change in consumer prices (with a one-period lag), thethe change in output and its level in the previous period. The model does not


distinguish between wages and the price of domestic output, so that thisis also the equation for domestic price inflation (y). Usually, domesticinflation is made proportional to the difference between demand andsupply for domestic output. Aggregate supply is in turn derived from aCobb-Douglas production function with capital fixed, labor demandresponding to the product wage and labor supply responding to the wagedeflated by the consumer price index

(7)

where c' is the share of capital in output and n is the labor supplyelasticity S

The author's formulation is, instead: C.

(8)

Equation (8) assumes that 0(1 —c')n = 1 (or, with c' = 0.5, a laborsupply elasticity of 6.7) since the real exchange rate elasticity is larger by i ¶

1/(1 + nc')when the product wage is fixed. More importantly, it incorporatesthe effect of cumulative output on inflation. If there is some inheriteddomestic price inflation, say y0, then it allows full employment, no inflationand no change in the real exchange rate in steady-state: V

(8')

The welfare effect of different policies refers only to the timing of outputlosses, given by Alternatively, even with = 0, as in Buiter andMiller (1982), the measure of output losses could be —(1or the change in the real exchange rate.

Using m as a policy variable, and ignoring the foreign variables, thesystem of(3), (4) and (8) has a block-triangular state-space representationin terms of p, r, y and q:

1 0 0 0 p 00 10 p Ô

0 1 0 0 r —0—1— r ——C C + em

0 —l+A 1 —0 j) 0 0 0

0 0 q 0

The system in (9) has one positive root, associated with the jump variabler, and three negative roots, as required for saddle-point stability. Theobjective function of the domestic authorities is expressed in terms of the inoutput gap and the rate of consumer price inflation:

• V..

U0 =J

+ çinT2) dt

Several numerical solutions of the maximization of (10) subject to (9)are presented in Section II of the paper, about two-thirds of the text.

A natural welfare ranking, based on the constraints to optimization, isgiven for the one-country model: open-loop control first, closed-loopcontrol second and time-consistent control third. Time-consistent control,

• though, does not rely on precommitments, so that the first best policy maynot be feasible. If the search is not confined to linear memoryless rules,so that the threat of reversion to the undesirable line consistent rule iscredible (because then the first best open-loop strategy canbe sustained.7 At the end of Section II, of the model with 0 = 0 in (8) isused to suggest that the utility difference between the first best rule andthe time consistent rule is given by (y—O.25 r)2/l000. For the 10%

• inherited inflation rate mentioned in the paper and r = 0, this wouldbe

The numerical rules presented before this example reflect the version ofthe model where 0 0. They seem to suggest that is superior towhich in turn is superior to mTC. This involves comparing the followingexpressions (we keep here the discrete time notation of the authors andtheir equation numbers preceded by a):

M?L = — 1.038 +0.257 qg_j—O.Ol 1 (a14)

= — 1.019 lTt+O.255 qg_1+O.389

= — 1.032 irg+0.258 (a24)

It is noted that and that in the early periods.The open-loop policy is overcontractionary (given the total output loss)

as far as future governments are concerned. While no ranking is given, theBuiter (1983) notion of time consistency (which does not allow thegovernment to optimize on the exchange rate and thus sets = 0)involves:

= —0.763 qg_1 (a29)

The time inconsistency of the rule is demonstrated by computing the firstperiod money stock under the Buiter criterion:

1y70BU = 1.147 0.287 r_1 + 0.309 q_1 (a30)

There is no guidance in the paper as to the interpretation of these results,in contrast with the experiments the authors run in the Brookings paper.

The research design seems to change in Section II, where the two-country

InternationaL coordination in dynamic macroeconomics 323

(10)

a

7)

'ly

8)

orby:es

edon

S')

(a25)

0,

he

n

,lehehe

j


model is solved analytically in a special case and the cooperative andnon-cooperative solutions are rooted in the size of the parameter & When

= = 0, then q = 6r and ffi = q in (2). This allows output to be controlleddirectly, as in Sachs (1983). It is also assumed that 0 = 0 in (8). Since

= —y, the system in this case reduces to one state-variable:

(ii)where A = (1 —A)/28

The Hamiltonian for the Nash non-cooperative solution is:

H = (12)

where ir (1 _A)y+Ay* and is the shadow price of domestic inflation.From the canonical equation for the costate variable and the first order

condition for a maximum, we obtain a differential equation for the controlvariable:

q = _A)y* (13)

Ignoring foreign inflation, (11) and (12) give a system with two roots(R) of opposite sign (and equal magnitude when /? = 0):

2R = fl±[(Ii+2A)2+4(1 (14)

The cooperative solution is obtained by setting r = 0 so that ir = y =This is equivalent to making A = 0 in (11) and The stable root islarger in this case so that, as we found in the steady-state example, thereis less of a contractionary bias under cooperation. The more efficient timingof inflation implies also a lower loss in this case. However, in the two-countryworld, it is not possible to rank policies because the open-loop is no longerfirst best. Indeed, even among time consistent policies, there is nopresumption in favor of cooperation which may be as overinflationary asthe open-loop non-cooperative is overdeflationary. This is, of course, oneof the basic insights drawn from early (but as yet largely unpublished)contributions by Canzoneri and Gray (1983), Eichengreen (1984), Cooper(1984), Rogoff (1983), Laskar (1984) and others.9

One is left hoping that a forthcoming Oudiz—Sachs paper will consolidatethe gains in insight by providing empirical evidence on the gains from (timeconsistent?) policy coordination in a multicountry model.

NOTES

I There are many references in this literature, which in a sense goes back toproject LINK. Aside from the Brookings paper, the asymmetry between the

- -

____________


Ld U.S. and other industrial countries or groups thereof is analyzed in Brandsma

.n and Hallett (1983) and Hallett (1984). There is also the exciting work of Taylor,say his (1984). Another deleted strand of literature assesses attitudes towardcooperation. See Axelrod (1984) and the survey of policy makers reported inDeane and Pringle (1984).

2 Solving the two-country IS model, we would obtain:

I) q = 8r_cr(i_1r)_o.*(i*_ir*)qt = _8r*_o.(i*_ir*)_o.(i_m)

• Note that the consumer price indices are used to deflate interest rates, sothat

2) q_q* = 26r+(o._o.*)(l _A_A*)

0It can be shown that a> If 1 —A = A* the effect disappears. The expected

negative effect stems from the transfer condition: A + A* > 1 if the exchange rateer does not enter, the coefficient becomes _(a_a*). See Macedo (1983a).

• ol 3 The LM curve would becomern_pc =

• 3) Using the definition of the consumer price index, we get the right hand sideof (2) as a cz-weighted average of q and (1 — A) r. The real exchange rate effect

S drops when = I. See, for example, Macedo (1984).4 This example, as discussed in Cooper (1982), may be counted as the first paper

4\ of Sachs on the subject. It represents a tutorial exposition of the beggar-/ thy-neighbor world of Canzoneri and Gray (1981). Along the same lines, Sachs

(1983) includes a traded intermediate input with weight a and a real price (s)fixed in terms of a basket with weight so that, if pV is the price of domesticvalue added, the consumer price index is given by:

rigpc = —a)s+O.5A(l —a)p+O.5(l —A)(l _a)(p*+e)

ry 5 This is shown in Macedo (1984).er 6 This is elaborated in Macedo (1983a and b). See also Rogoff (1983).

7 On reputation equilibria, see Backus and Drifill (l984a and b). A good survey

as of differential games is in Tirole (1982).8 This requires either that foreign prices do not enter the consumer price index

ie (A = 1) or that 8 be infinite. As mentioned, the implied trade elasticities wouldd) be implausible in a two-country model. In the first case, there would be noer relevant link between the two economies.

9 The classic contributions are Johnson on tariffs and retaliation, Niehans and

te Hamada, who dealt lastly with fiscal interdependence in a Diamond debt setup;see his (1984). In a sense, it all began with Hamada's application of game-theory

LW to the choice of the savings ratio in a two-country model of capital move-ments; see his (1965, 98—113). Cooper (one of his thesis advisers) and hisclassmate Bryant were encouraging. The field remained dormant for manyyears, until well after Bryant (1980). Thanks largely to the Sachs machine, itbegan to arouse attention in 1983.

tohe

a


REFERENCES(

Note: References in the paper, e.g. Buiter (1984), Buiter and Miller (1982), Oudizand Sachs (1984), Rogoff (1983) and Sachs (1983) are not repeated here. Ii

SiAxeirod, R. (1984). The Evolution of Cooperation. Basic Books.Backus, D. and J. Driffihl (1984a). 'Inflation and Reputation'. Institute for

Economic Research, Queen's University, Discussion Paper No. 560, ir

Feburary. e

(1984b). 'Rational Expectations and Policy Credibility Following a Change in 0:Regime'. Institute for Economic Research, Queen's University, Discussion p:Paper no. 564, June. tc

Brandsma, A. and A. Hallett (1984). 'Optimal Policies for Interdependent Econ-omies: Risk Aversion and the Problem of Information', in 1. Basar and L.Pau, eds., Dynamic Modelling and Control of National Economies, Pergamon I

Press. tOBryant, R. (1980), Money and Monetary Policy in Interdependent Economies.

Brookings Institution.Canzoneri, M. and J. Gray (1981). 'Monetary Policy Games Following an Oil Price

bShock', draft. Federal Reserve Board, December (revised as 'Monetary Policy a

Games and the Consequences of Non Cooperative Behavior', 1983).Cooper, R. (1984). 'Economic Interdependence and Coordination of Economic oil

Policies',in R. JonesandP. Kenen,eds., Handbookof International Economics.North Holland (first draft November 1982). (1

Deane, M. and R. Pringle (1984). Economic Cooperation from the Inside. NewYork, Group of Thirty.

Eichengreen, B. (1984). 'Central Bank Cooperation under the Interwar Gold re

Standard'. Explorations in Economic History. January (first draft January th1983). at

Hallett, A. (1984). 'Policy Design in Interdependent Economies: The Case forCoordinating US and EEC Policies', draft. Erasmus University, June.

thHamada, K. (1965). 'Economic Growth and long-term International CapitalMovement', Ph.D. dissertation, Yale University at

(1984). 'Strategic Aspects of International Fiscal Interdependence', draft. taUniversity of Tokyo, June. th

Laskar, D. (1984). 'Foreign Exchange Intervention Policies in a Two-Country selWorld: Optimum and Non-Cooperative Equilibrium', draft. CEPREMAP, cApril.

Macedo, J. (1983a). 'Policy Interdependence under Flexible Exchange Rates'.Woodrow Wilson School, Princeton University, Discussion Paper in Econ-omics No. 62, May.

(1983b). 'Small Countries in Monetary Unions: The Case of Senegal', draft.Princeton University, September.

(1984). 'Trade and Financial Interdependence under Flexible Exchange Rates:The Pacific Area', draft. Princeton University, August.

Taylor, J. (1984). 'International Coordination in the Design of MacroeconomicPolicy Rules', draft. Princeton University, June. ml

Tirole, J. (1982). 'Jeux Dynamiques: Un guide de l'utilisation'. CERAS Document andu Travail no. 7, June. cei

1k


COMMENT KENNETH ROGOFF*

Z In this remarkably clear-headed paper, Glues Oudiz and Jeff Sachs havesucceeded in both consolidating and extending the more technical game-theoretic literature on international monetary policy cooperation. Oneimportant contribution of their paper is to provide relatively simpleexamples of some difficult dynamic game theory concepts in the contextof a well-known international macro model. My specific comments on thepaper will deal with the question, raised here and in my 1983 paper, asto whether cooperation between central banks exacerbates or amelioratesthe credibility problem of the central banks vis-à-vis the private sector. But

L. I will also make some general remarks about applications of game theory)fl to international macroeconomics.

Until recently, the literature on international monetary cooperationfocused almost exclusively on the strategic interactions of sovereign central

- ce banks, each concerned solely with its own country's welfare. Because little• attention was paid to the problem of maintaining low time-consistent rates

ticof inflation, the strong presumption was that cooperation beween centralbanks in stabilization policy is unambiguously beneficial. In Rogoff(l983b), I formally demonstrated that this presumption was incorrect. Inthe absence of institutional constraints on systematic inflation, a cooperative

Idregime may quite possibly be characterized by higher mean inflation rates

ry than a noncooperative regime. Suppose, for example, that private agentsare concerned that the central bank will try to exploit the existence of

or nominal wage contracts to raise employment. Wage setters can frustratethe central bank by setting wage inflation high enough so that in the

al absence of disturbances, the central bank will choose to ratify wage setters'

rt. target real wage. At a high enough rate of inflation, the central bank findsthat the marginal gain from trying to lower employment below wage

ry setters' target level is offset by the marginal loss from still higher inflation.Consider how this inflationary bias may be exacerbated in a cooperativeregime. When central banks inflate jointly, none of them need worry abouthaving their real exchange rate depreciate. On the other hand, in anoncooperative regime, real depreciation provides an important check on

ft. each central bank's incentive to unanticipatedly expand its money supply,since depreciation lowers the employment gains and raises the inflation

S. costs. Because cooperation removes this check on the central banks'incentives to inflate, it raises the time-consistent level of nominal wageinflation. This basic result can be extended to alternative non-neutralities

• nt and alternative sources of time-consistency problems. Depending on thecentral banks' objective functions, it is also possible that the cooperative

328 Comment by Kenneth Rogoff

regime will be characterized by lower inflation. (The cooperative regime remay have lower time-consistent inflation rates if (a) the central banks'

WI

objective functions depend on employment and money supply growth (asin Canzoneri—Gray (1984)) rather than employment and inflation, and (b)the coefficients of the macro model are such that an unanticipated foreignmonetary expansion lowers employment at home.) Note also that in all ticases, cooperation produces superior responses to disturbances. B

Oudiz and Sachs have produced an example in which cooperationactually removes the source of the time-consistency problem, and isunambiguously welfare-improving. Their analysis seems particularly a 1

relevant to the disinflation problems faced by many countries over the pastten years. In the Oudiz—Sachs model, time-consistency problems arisebecause if a government unilaterally announces future right monetary apolicies, it causes its exchange rate to appreciate today thereby improving wits current-period Phillips curve trade-off. Since future governments willnot be concerned with how expectations of their policies affect today's hexchange rate, the optimal unilateral policy will not be time-consistent. thCooperation ensures that central banks will not try to manipulate the oexchange rate to their advantage and therefore removes this source of time oinconsistency. Note that there is no long-run systematic inflationary bias rain Oudiz and Sachs' model. En passant, it is worth mentioning that e

time-consistency problems can arise through a similar channel as in the pOudiz—Sachs paper in a closed-economy model in which both the price leveland the real interest rate enter the central bank's objective function.

I want to conclude with a few general remarks on the game theory Napproach to international macroeconomics embodied in many of thepapers in this conference. One issue is whether or not it is realistic tothe game between countries in monetary policy separately from the gamesinvolved in setting trade policy, defense policy, etc. There are certainlycases where countries tie commercial relations to defense relations: Andit is also not unusual for countries to simultaneously negotiate over tradeand monetary issues. The problem becomes especially acute when we R1

consider some of the dynamic concepts introduced by Oudiz and Sachs.Consider reputational equilibria, for example. If a country has recently Imisbehaved in trade policy, might that not affect its credibility in monetarypolicy? Secondly, I commend Oudiz and Sachs for reminding us in theirinteresting theoretical paper, that an important goal of this research shouldbe its ultimate application. In fact, Oudiz and Sachs' (l984a) Brookingspaper is one of the very few empirical papers in this literature. Matthew oCanzoneri has stressed in his comments here that empirical implementationappears difficult even for the Canzoneri—Gray (1984) model. In that R


erelatively simple model, one can deduce quite a lot simply by knowingwhether an unanticipated foreign money increase raises or lowers domesticemployment. Unfortunately, Canzoneri points out, the empirical evidenceon the sign of this effect is not decisive. In models with multiple

I instruments, and in which foreign policy affects multiple domestic objec-tives, the problem is even harder. For then, as in Oudiz and Sachs'Brookings paper, we need to know the magnitudes of all the multipliers,and not just their signs. It is also clear from analyses such as the present

is Oudiz—Sachs paper, that empirical implementation requires knowing quitea bit about the strategic behavior of the central banks. What type of gamethey are playing and with whom: the other country's central bank, their

— se own private sector and/or their own fiscal authorities? Clearly, it will be• ry a challenge to embody the new game-theoretic approach in future empirical

work.

ill Finally, we should recognize that game theory leads to thinking abouthow to modify institutions rather than policies. Hamada (1976) stressed

•that the problem of international monetary cooperation is best viewed as

he one in which countries cooperate to set up the system which works bestne on a day-to-day basis without cooperation. The new focus on institutionsas rather than just policies is a logical extension of the early work on rational

expectations, which emphasized the importance of analyzing systematiche policies rather than individual actions. (See Rogoff (1983a).)iel

ry NOTEhelel

* The views expressed here are those of the author and should not be interpreted

tes as reflecting the views of the Board of Governors of the Federal Reserve

ilySystem.

dewe REFERENCES

Canzoneri, Matthew B. and Jø Anna Gray (1984). 'Monetary policy games andEly the consequences of non-cooperative behavior'. International EconomicLry Review, forthcoming.eir Hamada, Koichi (1976). 'A strategic analysis of monetary interdependence'.ild Journal of Political Economy 84, Aug., 667—700.

Oudiz, Gilles and Jeffrey Sachs (l984a). 'Macroeconomic policy coordinationg among the industrial economies'. Brookingpapers on economic activity 1, 1—64.ew Oudiz, Gilles and Jeffrey Sachs (1984b). 'International policy coordination inOn dynamic macroeconomic models', this volume.iat Rogoff, Kenneth (1983a). 'The optimal degree of commitment to an intermediate

330 Comment by Kenneth Rogoff

monetary target: Inflation gains versus stabilization costs'. InternationalFinance Discussion Paper No. 230, Sept. (Board of Governors of the FederalReserve System, Washington, D.C.).

Rogoff, Kenneth (I 983b). 'Productive and counterproductivecooperative monetarypolicies.' International Finance Discussion Paper No. 233, December (Boardof Governors of the Federal Reserve System, Washington, D.C.).

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international policy coordination in dynamic macroeconomic

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